Well, considering the fact that most "exploits" are done first once gaining access to the network, I'm really failing to understand what it is you're trying to do.
However, if -say- you're trying test your firewall capability, you can easily try throwing commands at your IP address using a tunnel on various open ports. The idea is you want to try to figure out what it is the network is running, as many programs use specific ports and usually admins fail to use non-standard ones.
A good place to start, especially if you're running apache servers and such, is to constantly check for security holes in the updates for those products. Rule of thumb is to not necessarily stay up to date, but use stable builds.
I'm being quite vague in my post, I realize, but these are things that are easily looked up and are generally specific to a network.